Some effect of drift of the generalized Brownian motion process: Existence of the operator-valued generalized Feynman integral

Abstract

In this paper an analytic operator-valued generalized Feynman integral was studied on a very general Wiener space $C_{a,b}[0,T]$. The general Wiener space $C_{a,b}[0,T]$ is a function space which is induced by  the generalized Brownian motion process associated with continuous functions $a$ and $b$. The structure of the analytic operator-valued generalized Feynman integral is suggested and the existence of the analytic operator-valued generalized Feynman integral is investigated as an operator from $L^1(\mathbb R, \nu_{\delta,a})$ to $L^{\infty}(\mathbb R)$ where $\nu_{\delta,a}$ is a $\sigma$-finite measure on $\mathbb R$ given by
$$
d\nu_{\delta,a}=\exp\{\delta \mathrm{Var}(a)u^2\} du,
$$
where $\delta>0$ and $\mathrm{Var}(a)$ denotes the total variation of the mean function $a$ of the generalized Brownian motion process. It turns out in this paper that the analytic operator-valued generalized Feynman integrals of functionals defined by the stochastic Fourier--Stieltjes transform of complex measures on the infinite dimensional Hilbert space $C_{a,b}'[0,T]$ are elements of the linear space
$$
\bigcap_{\delta>0} \mathcal L( L^1(\mathbb R,\nu_{\delta,a}),L^{\infty}(\mathbb R)).
$$